2023 Fiscal Year Final Research Report

Global Study of Primitive Forms

Research Project

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Project/Area Number	18H01116
Research Category	Grant-in-Aid for Scientific Research (B)
Allocation Type	Single-year Grants
Section	一般
Review Section	Basic Section 11020:Geometry-related
Research Institution	Kyoto University (2019-2023) The University of Tokyo (2018)
Principal Investigator	Saito Kyoji 京都大学, 数理解析研究所, 名誉教授 (20012445)
Co-Investigator(Kenkyū-buntansha)	柏原正樹京都大学, 高等研究院, 特定教授 (60027381) 高橋篤史大阪大学, 大学院理学研究科, 教授 (50314290) 池田暁志城西大学, 理学部, 准教授 (40755162) 桑垣樹京都大学, 理学研究科, 准教授 (60814621) 社本陽太早稲田大学, 高等研究所, 講師(任期付) (50823647)
Project Period (FY)	2018-04-01 – 2023-03-31
Keywords	primitive form / elliptic root system / elliptic Lie algebra / elliptic Artin group / elliptic Artin monoid / hyperbolic root system / integrablerepresentation / cuspidal root system
Outline of Final Research Achievements	The program aims to construct global period map theory for primitive forms and its application to integrable systems. The following progresses were achieved during the planned period, but they were not completed and were succeeded in its following program to be published. (1) Books on the analytic theory of primitive forms and its application to integrable systems is prepared by a joint work. (2) Theory of sign decomposition for generalized root systems is constructed. As applications, we obtained (i) classification of hyperbolic and cuspidal root systems and (ii) highest weight integrable representations of the algebras associated with the generalized root systems beyond Kac-Moody theory. (3) Construction of elliptic Lie algebras and elliptic Artin groups together with modular group actions on them. (4) General construction of second homotopy classes for non-cancellative monoids, and its application to the elliptic Weyl group regular orbit spaces.
Free Research Field	complex geometry
Academic Significance and Societal Importance of the Research Achievements	原始型式の解析理論とその可積分系への応用の一般公開は長く求められていた。楕円リー環とその表現論はカッツ.ムーディリー環の枠を超えるもので、新たな幾何学や数理物理への応用が見込まれる。楕円周期領域が高次のホモトピー類を持つと言うことは Deligne による単体的アレンジメント補集合は$K(\pi,1)$-空間と言う従来の常識を超える新現象である。その事実が今後どの様な影響を持つのかは予測し難い。ハイパボリックないしカスピダルルート系の理論の完成はルート系の符号分解理論によるもので、その理論はカッツ.ムーディリー環の表現論を大幅に超える最高ウェイト表現論の建設を可能にし、応用研究が待たれる。